WASHINGTON — Unlikely allies, business and labor leaders joined in support of the White House’s immigration overhaul efforts Tuesday while also launching high-stakes negotiations to overcome an issue that has split them before — creating a guest-worker program to ensure future immigrants come to the U.S. legally.

The broad agreement on a need for immigration changes and a pathway to citizenship for an estimated 11 million illegal immigrants already here is driven largely by self-interest. Business and labor see an overhaul of the nation’s broken immigration system as a way to boost economic competitiveness with other nations while increasing the ranks of workers and union members.

For President Barack Obama, a partnership between factions that have often been at odds, with each other and with the White House, allows him to turn up pressure on Congress and try to isolate congressional Republicans who oppose parts of an immigration overhaul.

Obama held separate meetings at the White House on Tuesday with labor leaders and top business executives.

“This is all very encouraging to have labor and business come together to explore what could be some common ground,” said Janet Murguia, president of the National Council of La Raza, a leading immigration rights groups. Murguia and other immigration activists joined Obama’s meeting with labor groups.

Despite such optimistic public statements, the fragile business-labor alliance is still in question as the Chamber of Commerce meets with the AFL-CIO and other labor groups privately to hammer out details of how to deal with future immigrants who come to the U.S. to work.

The labor and business groups have been tasked by Sen. Charles Schumer, D-N.Y., with reaching a deal within weeks that can be included in legislation being crafted by a bipartisan Senate group, officials said.

The guest-worker issue helped scuttle the last attempt at a comprehensive overhaul of immigration law in 2007. If the parties can’t reach a deal, senators and their staffs are prepared to write temporary-worker language themselves, said a Senate aide, who spoke only on condition of anonymity because the aide was not authorized to discuss the private negotiations publicly.

The Senate negotiating group has included a guest-worker program in its immigration proposals, but Obama has not.

That omission has drawn criticism from Sen. Marco Rubio of Florida, a key negotiator on the Republican side. Republicans view the omission as a cave-in to labor supporters, who see a substantial new guest-worker program as a possible threat to Americans who are seeking jobs.

White House officials said the president is open to a guest-worker program, as long as it protects workers and responds to workforce demands, not politics.

That puts the White House in line with AFL-CIO President Richard Trumka, who said following his meeting with Obama Tuesday that they discussed “a data-driven system that is actually driven by needs and not by aspirations of employers.”